In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from the Latin stella, "star". Stellation is the reciprocal or dual process to faceting.
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the Bilinski dodecahedron. There are some stellations of the rhombic dodecahedron, one of which is the Escher's solid. The rhombic dodecahedron may also appear in the garnet crystal, the architectural philosophies, practical usages, and toys.
The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,5⁄2}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by Hess (1878), Badoureau (1881) and Pitsch (1882).
In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol b{5,5⁄2}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 5⁄3 5, and Coxeter diagram .
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.
The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.
There are two different compounds of great icosahedron and great stellated dodecahedron: one is a dual compound and a stellation of the great icosidodecahedron, the other is a stellation of the icosidodecahedron.
In geometry, faceting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.
In geometry, the first stellation of the rhombic dodecahedron is a self-intersecting polyhedron with 12 faces, each of which is a non-convex hexagon. It is a stellation of the rhombic dodecahedron and has the same outer shell and the same visual appearance as two other shapes: a solid, Escher's solid, with 48 triangular faces, and a polyhedral compound of three flattened octahedra with 24 overlapping triangular faces.
The expanded icosidodecahedron is a polyhedron, constructed as an expanded icosidodecahedron. It has 122 faces: 20 triangles, 60 squares, 12 pentagons, and 30 rhombs. The 120 vertices exist at two sets of 60, with a slightly different distance from its center.
